Quantum Painlevé systems of type $A_l^{(1)}$

Abstract

We propose quantum Painlevé systems of type $A_l^{(1)}$. These systems, for $l=1$ and $l\ge 2$, should be regarded as quantizations of the second Painlevé equation and the differential systems with the affine Weyl group symmetries of type $A_l^{(1)}$ studied by M. Noumi and Y. Yamada \cite{NYhigherorder}, respectively. These quantizations enjoy the affine Weyl group symmetries of type $A_l^{(1)}$ as well as the Lax representations. The quantized systems of type $A_1^{(1)}$ and type $A_l^{(1)}$ ($l=2n$) can be obtained as the continuous limits of the discrete systems constructed from the affine Weyl group symmetries of type $A_2^{(1)}$ and $A_{l+1}^{(1)}$, respectively.

26 pages; v2, minor changes, corrected typos, add $l=1$ case to Section 2

References in corpus (3)

• Higher order Painlevé equations of type $A_l^{(1)}$
• A study on the fourth q-Painlevé equation
• Affine Weyl group approach to Painlevé equations

Cited by in corpus (6)

• Non-commutative lattice modified Gel'fand-Dikii systems
• Quantum Painlevé Equations: from Continuous to Discrete
• On the Tracy-Widom${}_{\beta }$ Distribution for $\beta =6$
• Quasideterminant solutions of NC Painlevé II equation with the Toda solution at $n=1$ as a seed solution in its Darboux transformation
• Quantum groups and quantization of Weyl group symmetries of Painlevé systems
• Takano's Theory of Quantum Painleve Equations